The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.

Theorem. If $A$ is an invertible upper triangular matrix, then $A^{-1}$ is also upper triangular. Similarly, if $A$ is an invertible lower triangular matrix, then $A^{-1}$ is lower triangular.

Note: A triangular matrix is invertible if and only if all its diagonal entries are nonzero.

Proof idea. For an upper triangular matrix $A$, the inverse can be computed by back-substitution, which preserves the upper triangular structure.

Example. The inverse of $A = \begin{pmatrix} 2 & 3 \\ 0 & 4 \end{pmatrix}$ is $A^{-1} = \begin{pmatrix} 1/2 & -3/8 \\ 0 & 1/4 \end{pmatrix}$, which is also upper triangular.

The diagonal entries of $A^{-1}$ are the reciprocals of the diagonal entries of $A$: $(A^{-1})_{ii} = 1/a_{ii}$.